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| Peridynamics is a model in Solid Mechanics formulated by S. Silling in 2000.
Its main difference with the usual Cauchy--Green elasticity relies in its non-locality, which reflects the fact that particles at a positive distance exert an interaction force upon each other.
Mathematically, the deformations are not assumed to have weak derivatives, in contrast with classical Continuum Mechanics, and in particular, hyperelasticity, where they are assumed to be Sobolev.
This makes peridynamics a suitable framework for mechanical problems where discontinuities appear naturally, such as fracture, dislocation or general multiscale materials.
In this talk we study the variational theory of time-independent problems in peridynamics.
In this formulation, the energy of a deformation is expressed as a double integral, and resembles the energy functional of a nonlocal singular $p$-Laplacian.
In particular, we will explain optimal conditions over the integrand guaranteeing that the energy admits a minimum, and, hence, that the problem has an equilibrium solution.
Moreover, we will see how the Cauchy--Green elasticity theory can be recovered from this nonlocal theory as a $\Gamma$-limit passage when the interaction distance tends to zero. |
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